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30
Some Connections between Bounded Query Classes and NonUniform Complexity
 In Proceedings of the 5th Structure in Complexity Theory Conference
, 1990
"... This paper is dedicated to the memory of Ronald V. Book, 19371997. ..."
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Cited by 70 (23 self)
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This paper is dedicated to the memory of Ronald V. Book, 19371997.
PolynomialTime Membership Comparable Sets
, 1994
"... This paper studies a notion called polynomialtime membership comparable sets. For a function g, a set A is polynomialtime gmembership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m j ..."
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Cited by 31 (4 self)
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This paper studies a notion called polynomialtime membership comparable sets. For a function g, a set A is polynomialtime gmembership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m jg), outputs b 2 f0; 1g m such that (A(x 1 ); \Delta \Delta \Delta ; A(xm )) 6= b. The following is a list of major results proven in the paper. 1. Polynomialtime membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomialtime membership comparable sets have polynomialsize circuits. 3. For any function f and for any constant c ? 0, if a set is p f(n)tt reducible to a Pselective set, then the set is polynomialtime (1 + c) log f(n)membership comparable. 4. For any C chosen from fPSPACE;UP;FewP;NP;C=P;PP;MOD 2 P; MOD 3 P; \Delta \Delta \Deltag, if C ` Pmc(c log n) for some c ! 1, then C = P. As a corollary of the last tw...
ResourceBounded Kolmogorov Complexity Revisited
 In Proceedings of the 14th Symposium on Theoretical Aspects of Computer Science
, 2001
"... We take a fresh look at CD complexity, where CD (x) is the size of the smallest program that distinguishes x from all other strings in time t(jxj). We also look at CND complexity, a new nondeterministic variant of CD complexity, and timebounded Kolmogorov complexity, denoted by C complexity. ..."
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Cited by 25 (7 self)
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We take a fresh look at CD complexity, where CD (x) is the size of the smallest program that distinguishes x from all other strings in time t(jxj). We also look at CND complexity, a new nondeterministic variant of CD complexity, and timebounded Kolmogorov complexity, denoted by C complexity.
Optimal Advice
 Theoretical Computer Science
, 1994
"... Ko [Ko83] proved that the Pselective sets are in the advice class P/quadratic. We prove that the Pselective sets are in NP=linear T coNP=linear. We show this to be optimal in terms of the amount of advice needed. 1 Introduction Selective sets are sets for which there is a "selector function," u ..."
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Cited by 15 (12 self)
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Ko [Ko83] proved that the Pselective sets are in the advice class P/quadratic. We prove that the Pselective sets are in NP=linear T coNP=linear. We show this to be optimal in terms of the amount of advice needed. 1 Introduction Selective sets are sets for which there is a "selector function," usually a polynomialtime deterministic or nondeterministic function, that selects which of any two given inputs is logically no less likely than the other to belong to the given set. Definition 1.1 [HNOS94] Let FC be any class of functions (possibly multivalued or partial). A set A is FCselective if there is a function f 2 FC such that for every x and y, it holds that f(x; y) ` fx; yg, and if fx; yg " A 6= ;, then f(x; y) 6= ; and f(x; y) ` A. Let FCsel denote the class of sets that are FCselective. The class that would be notated FP single\Gammavalued; total sel according to the definition above was defined directly by Selman in 1979 [Sel79]. Henceforward, we refer to these sets ...
Competing Provers Yield Improved KarpLipton Collapse Results
 Information and Computation
, 2002
"... Via competing provers, we show that if a language A is selfreducible and has polynomialsize circuits then S 2 = S 2 . Building on this, we strengthen the Kamper AFK Theorem, namely, we prove that if NP coNP)/poly then the polynomial hierarchy collapses to S 2 . We also strengthen Yap ..."
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Cited by 15 (2 self)
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Via competing provers, we show that if a language A is selfreducible and has polynomialsize circuits then S 2 = S 2 . Building on this, we strengthen the Kamper AFK Theorem, namely, we prove that if NP coNP)/poly then the polynomial hierarchy collapses to S 2 . We also strengthen Yap's Theorem, namely, we prove that if NP coNP/poly then the polynomial hierarchy collapses to S 2 . Under the same assumptions, the best previously known collapses were to ZPP respectively ([KW98, BCK 94], building on [KL80, AFK89, Kam91, Yap83]). It is known that S 2 [Cai01]. That result and its relativized version show that our new collapses indeed improve the previously known results. Since the Kamper AFK Theorem and Yap's Theorem are used in the literature as bridges in a variety of resultsranging from the study of unique solutions to issues of approximationour results implicitly strengthen all those results.
Reductions between Disjoint NPPairs
 Information and Computation
, 2004
"... We prove that all of the following assertions are equivalent: There is a manyone complete disjoint NPpair; there is a strongly manyone complete disjoint NPpair; there is a Turing complete disjoint NPpair such that all reductions are smart reductions; there is a complete disjoint NPpair for one ..."
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Cited by 15 (4 self)
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We prove that all of the following assertions are equivalent: There is a manyone complete disjoint NPpair; there is a strongly manyone complete disjoint NPpair; there is a Turing complete disjoint NPpair such that all reductions are smart reductions; there is a complete disjoint NPpair for onetoone, invertible reductions; the class of all disjoint NPpairs is uniformly enumerable. Let A, B, C, and D be nonempty sets belonging to NP. A smart reduction between the disjoint NPpairs (A, B) and (C, D) is a Turing reduction with the additional property that if D. We prove under the reasonable assumption UP coUP has a Pbiimmune set that there exist disjoint NPpairs (A, B) and (C, D) such that (A, B) is truthtable reducible to (C, D), but there is no smart reduction between them. This paper contains several additional separations of reductions between disjoint NPpairs. We exhibit an oracle relative to which DisjNP has a truthtablecomplete disjoint NPpair, but has no manyonecomplete disjoint NPpair.
SemiMembership Algorithms: Some Recent Advances
 SIGACT News
, 1994
"... A semimembership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A. A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent halfdecade. However, in the 1990 ..."
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Cited by 12 (8 self)
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A semimembership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A. A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent halfdecade. However, in the 1990s there has been a resurgence of interest in this topic. We survey recent work on the theory of semimembership algorithms. 1 Introduction A membership algorithm M for a set A takes as its input any string x and decides whether x 2 A. Informally, a semimembership algorithm M for a set A takes as its input any strings x and y and decides which is "no less likely" to belong to A in the sense that if exactly one of the strings is in A, then M outputs that one string. Semimembership algorithms have been studied in a number of settings. Recursive semimembership algorithms (and the associated semirecursive setsthose sets having recursive semimembership algorithms) were introduced in the 1...
Oracles That Compute Values
, 1997
"... . This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomialtime reducibilities, it is shown ..."
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Cited by 11 (4 self)
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. This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomialtime reducibilities, it is shown that 1. A multivalued partial function is polynomialtime computable with k adaptive queries to NPMV if and only if it is polynomialtime computable via 2 k \Gamma 1 nonadaptive queries to NPMV. 2. A characteristic function is polynomialtime computable with k adaptive queries to NPMV if and only if it is polynomialtime computable with k adaptive queries to NP. 3. Unless the Boolean hierarchy collapses, for every k, k adaptive (nonadaptive) queries to NPMV is different than k + 1 adaptive (nonadaptive) queries to NPMV. Nondeterministic reducibilities, lowness and the difference hierarchy over NPMV are also studied. The difference hierarchy for partial functions does not collapse unless the Boolean hierarchy collapses, but, surprisingly, the levels of the difference and bounded query hierarchies do not interleave (as is the case for sets) unless the polynomial hierarchy collapses. Key words. computational complexity, complexity classes, relativized computation, bounded query classes, Boolean hierarchy, multivalued functions, NPMV AMS subject classifications. 68Q05, 68Q10, 68Q15, 03D10, 03D15 1.
On the Structure of Low Sets
 PROC. 10TH STRUCTURE IN COMPLEXITY THEORY CONFERENCE, IEEE
, 1995
"... Over a decade ago, Schöning introduced the concept of lowness into structural complexity theory. Since then a large body of results has been obtained classifying various complexity classes according to their lowness properties. In this paper we highlight some of the more recent advances on selected ..."
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Cited by 10 (2 self)
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Over a decade ago, Schöning introduced the concept of lowness into structural complexity theory. Since then a large body of results has been obtained classifying various complexity classes according to their lowness properties. In this paper we highlight some of the more recent advances on selected topics in the area. Among the lowness properties we consider are polynomialsize circuit complexity, membership comparability, approximability, selectivity, and cheatability. Furthermore, we review some of the recent results concerning lowness for counting classes.